Subgroups — #88945

Order 432.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊#157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#157	C₂	1	1	yes	maximal, characteristic
4	#3777	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃⋊#157	—	2	1	no	—
7	C₃⋊#157	C₄	1	2	yes	complemented
8	C₃×#157	—	2	1	no	—
9	C₂×#157	—	3	1	no	—
10	C₂×#157	—	3	1	no	—
11	C₂×#157	D₃	1	1	yes	characteristic
12	C₂×C₃³	Q₈	1	1	yes	characteristic
13	C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
14	C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
15	Dic₃⋊C₄	—	9	1	no	maximal
16	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
17	C₂×C₃×D₃	—	12	1	no	—
18	#157	—	2	1	no	—
19	#157	—	3	2	no	—
20	#157	—	6	1	no	—
21	C₃³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	9	1	no	—
23	C₂×Dic₃	—	9	1	no	—
24	C₂×Dic₃	—	9	1	no	—
25	C₂×C₃²	—	4	1	no	—
26	C₂×C₃²	—	8	1	no	—
27	C₂×C₃²	Dic₆	1	1	yes	characteristic
28	C₃⋊D₃	C₄×D₃	1	1	yes	characteristic
29	C₃⋊D₃	#67	1	1	yes	characteristic
30	C₃×D₃	—	12	1	no	—
31	C₃×D₃	—	12	1	no	—
32	C₄⋊C₄	—	27	1	no	Sylow
33	C₂²×C₃	—	9	1	no	—
34	C₂×D₃	—	12	1	no	—
35	C₃×C₄	—	18	1	no	—
36	Dic₃	—	9	2	no	—
37	Dic₃	—	18	1	no	—
38	C₃²	—	4	1	no	—
39	C₃²	—	8	1	no	—
40	C₃²	Dic₃⋊C₄	1	1	yes	characteristic, complemented
41	C₂×C₄	—	9	1	no	—
42	C₂×C₄	—	27	1	no	—
43	C₂×C₄	—	27	1	no	—
44	C₂×C₃	—	4	1	no	—
45	C₂×C₃	—	8	1	no	—
46	C₂×C₃	—	9	1	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic
49	D₃	—	12	1	no	—
50	D₃	—	12	1	no	—
51	C₂²	—	9	1	no	—
52	C₄	—	18	1	no	—
53	C₄	—	27	2	no	—
54	C₄	—	54	1	no	—
55	C₃	—	4	1	no	—
56	C₃	—	8	1	no	—
57	C₃	#3777	1	1	yes	characteristic, complemented
58	C₂	—	9	1	no	—
59	C₂	—	9	1	no	—
60	C₂	C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊PSU(3,𝔽₂)
commutator	C₃×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃³	Q₈
prime radical, cosocle	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃³	C₄⋊C₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₃⋊#157	C₄	semidirect
21. C₃³	C₄⋊C₄	semidirect
40. C₃²	Dic₃⋊C₄	semidirect
57. C₃	#3777	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃⋊#157	2
C₂×C₃×#157	2
#3777	3
Dic₃⋊C₄	9